SPH4U Dynamics, Fields, and Modern Physics Exam Review Notes

Kinematics and Dynamics

• One-Dimensional Braking Analysis:

  • Scenario: A motorcyclist is travelling at a velocity of $15.0\,m/s$ [forward] and applies the brakes.

  • Acceleration: The motorcycle slows down at a rate of $5.0\,m/s^2$ [backward].

  • Objective: Determine the motorcycle’s braking distance.

• Linear Sprinting Kinematics (100.0 m Sprint):

  • Initial State: The runner starts from rest ($v_1 = 0\,m/s$).

  • Acceleration Phase: The runner reaches a velocity of $9.6\,m/s$ [W] in a time interval of $4.2\,s$.

  • Tasks:

    • Calculate the acceleration of the runner during the initial phase.

    • Calculate the displacement of the runner during this acceleration phase.

    • The runner maintains a constant velocity for the remainder of the race. Calculate the total time required to complete the $100.0\,m$ distance.

• Two-Dimensional Vector Displacement:

  • Leg 1: A person drives $120\,km$ [N 32\degree W] to a friend’s location.

  • Leg 2: The person then drives $150\,km$ [W 24\degree N] to visit family.

  • Objective: Determine the total displacement of the entire trip, accounting for both magnitude and direction.

• Horizontal Projectile Motion (Marble on a Table):

  • Initial Conditions: A marble rolls off a table with a horizontal velocity of $1.93\,m/s$.

  • Vertical Dimensions: The tabletop height is $76.5\,cm$ ($0.765\,m$) above the floor.

  • Assumptions: Air resistance is negligible.

  • Tasks:

    • Determine the duration (time) the marble is in the air.

    • Calculate the horizontal distance (range) the marble travels before striking the floor.

    • Calculate the final velocity of the marble at the moment of impact.

• Angled Projectile Motion (Baseball Toss):

  • Launch Parameters: A baseball is tossed from a second-floor window with an initial velocity of $4.3\,m/s$ at an angle of $42\degree$ above the horizontal.

  • Initial Height: The ball starts at a vertical position of $3.9\,m$.

  • Target Height: The ball is caught at a height of $1.4\,m$ above the ground.

  • Tasks:

    • Calculate the total time the ball remains in the air.

    • Calculate the horizontal distance between the window and the point where the ball is caught.

    • Calculate the maximum height reached by the ball relative to the ground.

    • Calculate the speed of the ball at the instant it is caught.

• Relative Velocity (Ferry Boat):

  • Frame of Reference: The ship is moving forward with a velocity of $2.8\,m/s$ relative to the water.

  • Object Motion: A group of people walks on the deck with a velocity of $1.1\,m/s$ relative to the deck.

  • Scenarios:

    • Determine the group's velocity relative to the water when walking toward the front of the ship.

    • Determine the group's velocity relative to the water when walking toward the back of the ship.

• Relative Velocity (Aviation):

  • Airspeed: A plane flies with a velocity relative to the air of $3.5 \times 10^2\,km/h$ [N 35\degree W] over Hamilton.

  • Wind Velocity: The wind is blowing at $62\,km/h$ [S].

  • Objective: Determine the resultant velocity of the plane relative to the ground.

• Relative Velocity (River Crossing):

  • Channel Width: The river is $84\,m$ wide.

  • Current: The water moves with a velocity of $0.40\,m/s$ [E].

  • Swimmer speed: The person swims at $0.70\,m/s$ [N] relative to the water.

  • Tasks:

    • Calculate the time required to cross the river.

    • Calculate the downstream distance the person will land from their starting longitude.

    • Determine the heading (direction) the person should swim to land at a point directly north of the starting position.

• Dynamics on an Inclined Plane with Friction:

  • Scenario: A sled takes off from the top of a hill inclined at $6\degree$ to the horizontal.

  • Initial Velocity: $12\,m/s$.

  • Friction: The coefficient of kinetic friction ($\mu_k$) between the sled and snow is $0.14$.

  • Objective: Determine the total sliding distance before the sled comes to a rest.

• Connected Systems (Atwood-style Table Machine):

  • Block A: Positioned on a level table with a mass of $12\,kg$.

  • Block B: Hanging from a pulley over the table edge with a mass of $6\,kg$.

  • Friction: The coefficient of friction between Block A and the table surface is $0.1$.

  • Tasks: Calculate the acceleration of the system and the magnitude of the tension in the connecting cable.

• Frictionless Inclined Plane:

  • Mass: $10\,kg$.

  • Angle: The incline is $22\degree$ above the horizontal.

  • Objective: Determine the acceleration of the block down the plane.

Circular Motion

• Horizontal Circular Kinematics:

  • Context: A ball on a string moves in a horizontal circle.

  • Radius: $1.4\,m$.

  • Centripetal Acceleration: Magnitude of $12\,m/s^2$.

  • Objective: Calculate the speed of the ball.

• Rodeo Rope Rotation:

  • Context: A performer twirls a rope at a constant speed.

  • Radius of Circle: $0.42\,m$.

  • Period ($T$): $1.5\,s$.

  • Objective: Determine the magnitude of the centripetal acceleration.

• Tension in Horizontal Circular Motion:

  • Mass: $2.00\,kg$.

  • System: Spinning horizontally on a frictionless surface, attached to a string $4.00\,m$ long.

  • Frequency/Period: Completes $5.00$ revolutions in $2.00\,s$.

  • Objective: Calculate the magnitude of the tension in the string, neglecting air resistance.

Energy and Momentum

• Work Done by Lifting:

  • Force: $275\,N$ exerted directly upward.

  • Displacement: $0.65\,m$.

  • Objective: Determine the work done on the weights by the weightlifter.

• Work and Displacement:

  • Scenario: Pushing on a wall with a constant force of $9.4\,N$.

  • Displacement: The wall does not move ($\Delta d = 0$).

  • Objective: Calculate the work done on the wall.

• Work Done at an Angle with Friction:

  • Sled Pull: A hiker pulls a sled over a distance of $223\,m$ using a constant force of $122\,N$ at an angle of $37\degree$ relative to the displacement.

  • Resistance: Friction acts on the sled with a constant force of $72.3\,N$.

  • Objective: Calculate the work done on the sled by the hiker and the work done by friction.

• Work-Energy Theorem (Stopping Distance):

  • Motion: A skater moves across ice for a distance of $12\,m$.

  • Braking: A constant frictional force of $15\,N$ causes the skater to stop.

  • Initial Speed: $2.2\,m/s$.

  • Objective: Calculate the mass of the skater.

• Conservation of Energy (Soccer Ball):

  • Mass: $0.43\,kg$.

  • Incline: A smooth frictionless hill $18\,m$ in height.

  • Initial Speed: $7.4\,m/s$.

  • Objective: Calculate the ball’s speed upon reaching the bottom of the hill.

• Spring Dynamics:

  • Configuration: A $5.3\,kg$ mass hangs vertically from a spring.

  • Spring Constant ($k$): $720\,N/m$.

  • Action: The mass is lifted upward and released.

  • Objective: Calculate the force and the acceleration on the mass at the moment the spring is compressed by $0.36\,m$.

• Elastic Potential Energy:

  • Device: A spring-loaded toy fires a marble.

  • Loading: A force of $220\,N$ is used to compress the spring by $0.14\,m$.

  • Objective: Calculate the elastic potential energy ($U_e$) stored in the toy.

• Conservation of Momentum (1D Explosion):

  • Scenario: Two stationary hockey players push off each other and move in opposite directions.

  • Player 1: Mass of $36\,kg$ and a speed of $3.3\,m/s$.

  • Player 2: Speed of $2.4\,m/s$.

  • Objective: Calculate the mass of the second player.

• Completely Inelastic Collision (2D):

  • System: Two trains collide at a track crossing.

  • Engine 1: Mass $14000\,kg$, initial velocity $45\,km/h$ [N].

  • Engine 2: Mass $15000\,kg$, initial velocity $53\,km/h$ [W].

  • Objective: Calculate the final velocity of the coupled engines.

• Elastic/Glancing Collision (2D):

  • Object 1 (Puck): Mass $0.16\,kg$, initial velocity $2.0\,m/s$ [E].

  • Object 2 (Puck): Mass $0.17\,kg$, initially at rest.

  • Post-Collision: The first puck has a velocity of $1.5\,m/s$ [N $31\degree$ E].

  • Objective: Determine the final velocity (magnitude and direction) of the second puck.

Gravitational, Electric, and Magnetic Fields

• Newton's Law of Universal Gravitation:

  • Masses: $m_1 = 1.0 \times 10^{20}\,kg$ and $m_2 = 3.0 \times 10^{20}\,kg$.

  • Force: Gravitational attraction is $2.2 \times 10^9\,N$.

  • Objective: Calculate the separation distance between the two asteroids.

• Gravitational Field Strength on Titan:

  • Field Magnitude ($g$): $1.3\,N/kg$.

  • Mass of Moon: $1.3 \times 10^{23}\,kg$.

  • Objective: Calculate the radius of Titan.

• Satellite Orbital Mechanics:

  • Orbit: Circular orbit at an altitude of $600\,km$ above Earth’s surface.

  • Tasks: Calculate the orbital speed and the orbital period of the satellite in minutes.

• Geosynchronous Orbit:

  • Objective: Calculate the orbital radius required for a satellite to remain in a geosynchronous orbit.

• Electrostatic Force (Coulomb's Law):

  • Charges: $q_1 = 1.0 \times 10^{-4}\,C$ and $q_2 = 1.00 \times 10^{-5}\,C$.

  • Separation: $2.0\,m$.

  • Objective: Determine the magnitude of the electric force between the charges.

• Electrostatic Force Superposition:

  • Charge 1: $+2.0 \times 10^{-6}\,C$ at $x = 0$.

  • Charge 2: $-3.0 \times 10^{-6}\,C$ at $x = 40.0\,cm$.

  • Charge 3: $-5.0 \times 10^{-6}\,km$ at $x = 120.0\,cm$ (Note: Transcript uses the unit 'km' for charge 3).

  • Objective: Determine the total net force acting on the $-3.0 \times 10^{-6}\,C$ charge.

• Electric Field Strength:

  • Source: Positive point charge $q = 6.25 \times 10^{-6}\,C$.

  • Point of Interest: $2.50\,m$ to the right of the charge.

  • Objective: Calculate the magnitude and direction of the electric field.

• Electrostatic Potential Energy and Particle Acceleration:

  • System: Two electrons start from rest separated by $5.0 \times 10^{-12}\,m$.

  • Action: Electrons are released and accelerate due to mutual repulsion.

  • Objective: Calculate the final speed of each electron when they are an infinite (very large) distance apart.

• Magnetic Field Mapping:

  • Conductors: Sketch magnetic fields for straight, current-carrying conductors and indicate direction.

  • B-Field Analysis: Determine the direction of the current creating specific magnetic field patterns.

  • Solenoids: Label the North pole of solenoids based on current flow (+ to - terminals).

• Magnetic Force on Moving Charges and Wires:

  • Proton in B-field: Mass $1.67 \times 10^{-27}\,kg$, moving horizontally eastward at $9.4 \times 10^4\,m/s$ into a $1.8\,T$ field directed vertically upward. Calculate magnitude and direction of the force.

  • Current in Truck Motor: Force of $1.4 \times 10^{-5}\,N$ on a $0.045\,m$ wire segment. Angle is $18\degree$ to a field of $5.3 \times 10^{-5}\,T$. Calculate the current.

  • Circular Path in B-field: Proton ($m = 1.67 \times 10^{-27}\,kg$) moves in a circle ($r = 8.0\,cm$) perpendicular to a $1.5\,T$ field. Calculate the velocity.

The Wave Nature of Light

• Interference Concepts:

  • Constructive Interference: Occurs when waves meet in phase (crest to crest), resulting in a combined wave with a larger amplitude.

  • Destructive Interference: Occurs when waves meet out of phase (crest to trough), resulting in a combined wave with a smaller or zero amplitude.

• Two-Point Source Interference:

  • Parameters: Sources vibrate in phase, distance $d = 3.0\,m$, wavelength $\lambda = 0.47\,m$.

  • Objective: Determine the angle to the 3rd nodal line.

• Scientific History of Light:

  • Requirement: Describe contributions of three scientists to the accepted modern model of light.

• Young’s Double Slit Experiment:

  • Description: Explain the experimental setup and the resulting interference pattern.

  • Significance: Discuss how this experiment provided evidence for the wave nature of light.

• Interference and Diffraction Calculations:

  • Double Slit I: Fifth-order dark fringe at $3.8\degree$ with slit separation $0.042\,mm$. Calculate $\lambda$.

  • Double Slit II: Second-order dark fringe of $650\,nm$ light with slit distance $6.3 \times 10^{-6}\,m$. Find the angle.

  • Thin-Film Interference: Calculate smallest thickness of a soap film ($n = 1.35$) on glass ($n = 1.52$) for reflective destructive interference with $\lambda = 745\,nm$.

  • Single Slit Diffraction: Slit width $3.00 \times 10^6\,m$ (per transcript). Angle between first dark fringes is $25.0\degree$. Calculate $\lambda$.

  • Diffraction Grating: Third-order bright fringe at $22\degree$ for red light ($\lambda = 694.3\,nm$). Calculate lines per centimetre.

Revolutions in Modern Physics

• Time Dilation:

  • Scenario 1: Clock moving at $0.60c$. Calculate how much longer a $1.00$ proper time interval appears to a stationary observer.

  • Scenario 2: An $8.0\,s$ interval on a moving spacecraft is measured as $10.0\,s$ on Earth. Calculate the relative speed ($v$).

• Length Contraction:

  • Parameters: Spacecraft 1 passes spacecraft 2 at $0.755c$. Observer on spacecraft 1 measures spacecraft 2 as $475\,m$ long.

  • Objective: Calculate the proper length of spacecraft 2.

• Relativistic Energy:

  • Particle: Proton moving at speed $0.800c$.

  • Tasks: Calculate the total energy and the kinetic energy ($E_k$) of the proton in mega-electronvolts ($MeV$).

• Photoelectric Effect:

  • Work Function ($W$): $5.0\,eV$.

  • Objective: Determine the minimum photon frequency (threshold frequency) required to eject an electron from the metal surface.